====== Convolution ======

[[http://24.229.196.141:81/|Not that convolution...]]

===== Definition =====

Discrete time convolution:

$$ (f \ast g)[n] \equiv \sum_{m = -\infty}^{\infty} f[m]g[n-m] $$

Continuous time convolution:

$$ (f \ast g)(t) \equiv \int_{-\infty}^{\infty} f(\tau)g(t - \tau) d\tau $$

===== Properties =====

  * Convolution is commutative:

$$ f \ast g = g \ast f $$

  * Convolution is associative:

$$ f \ast (g \ast h) = (f \ast g) \ast h $$

  * Convolution is distributive over addition:

$$ (f + g) \ast x = f \ast x + g \ast x $$
===== Frequency domain =====

  * Convolution in the time/spatial domain is equivalent to multiplication in the frequency domain. The inverse is also true.